Finite Group Databasebeta

← #4933818

Subgroups — #4933818

Order 512.

3246 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2431C211yesmaximal, characteristic
3#16559C212yesmaximal, complemented
4#16357C214yesmaximal, complemented
5#15884C212yesmaximal, complemented
6#15674C212yesmaximal, complemented
7#15468C214yesmaximal, complemented
8#15377C214yesmaximal, complemented
9#15050C212yesmaximal, complemented
10#14737C214yesmaximal, complemented
11#14419C212yesmaximal, complemented
12#13956C212yesmaximal, complemented
13#13832C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#121C2211yescharacteristic
18#2804—28no—
19#2431—28no—
20C2×#400C2212yes—
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#393C2214yes—
24C2×#393C2214yes—
25C2×#393C2212yes—
26C2×#392C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#381C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2212yes—
42#1446C2214yes—
43#1446C2212yes—
44#1445C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2212yes—
62#1436C2212yes—
63#1434C2212yes—
64#1434C2214yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2312yes—
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#127—24no—
86C2×#127—28no—
87C2×#124—24no—
88C2×#124—44no—
89C2×#121—24no—
90C2×#121—42no—
91C2×#121—42no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#400—28no—
113#397—44no—
114#397—44no—
115#393—216no—
116#393—48no—
117#393—44no—
118#392—216no—
119#391—24no—
120#391—28no—
121#390—44no—
122#390D418yes—
123#386—28no—
124#386—28no—
125#386—216no—
126#386—48no—
127#386—48no—
128#386—44no—
129#386—44no—
130#385—216no—
131#381—44no—
132C4×C22⋊C4—24no—
133C2×#95—44no—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C25C2412yes—
159C2×#41—416no—
160C2×#41—432no—
161C2×#41—432no—
162C2×#41—416no—
163C2×#41—416no—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—28no—
187C22×D4—42no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D414yes—
202C22×D4#4118yes—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235#127—416no—
236#124—416no—
237#121—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D4116yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—42no—
316#95—42no—
317C24—22no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24C22×D414yes—
342C24C2×#4114yes—
343C24C2×#4114yes—
344C24H(5,𝔽2)12yes—
345C24C2511yescharacteristic
346#41—4128no—
347C2×Q8—28no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—88no—
356C2×Q8—88no—
357C2×D4—216no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4#124116yes—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)14yes—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—44no—
486C42—44no—
487C42—84no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—41no—
522C23—42no—
523C23—44no—
524C23—41no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23C2×#12418yes—
548C23C23×D411yescharacteristic
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)12yes—
553C23C2×H(5,𝔽2)12yes—
554C23#58514yes—
555C23#58512yes—
556Q8—416no—
557Q8—816no—
558D4—416no—
559D4—432no—
560D4—88no—
561D4—832no—
562D4—816no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—416no—
580C2×C4—432no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—816no—
625C2×C4—816no—
626C2×C4C2×#12414yes—
627C22—216no—
628C22—28no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—48no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—48no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22C22×#12411yescharacteristic
656C22#348411yescharacteristic
657C22#348411yescharacteristic
658C22#348512yes—
659C22#348812yes—
660C22#349012yes—
661C22#349012yes—
662C22#351314yes—
663C22#352114yes—
664C22#352911yescharacteristic
665C22#353312yes—
666C22#354012yes—
667C22#354614yes—
668C22#354911yescharacteristic
669C22#355812yes—
670C22#356214yes—
671C4—24no—
672C4—44no—
673C4—42no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—84no—
678C4—84no—
679C4—84no—
680C4—84no—
681C2—48no—
682C2—82no—
683C2—162no—
684C2#3536711yescharacteristic
685C2#3584712yes—
686C2#4573012yes—
687C2#4622511yescharacteristic
688C2#4955112yes—
689C2#4963812yes—
690C2#5010414yes—
691C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2804C2semidirect
3. #16559C2semidirect
4. #16357C2semidirect
5. #15884C2semidirect
6. #15674C2semidirect
7. #15468C2semidirect
8. #15377C2semidirect
9. #15050C2semidirect
10. #14737C2semidirect
11. #14419C2semidirect
12. #13956C2semidirect
13. #13832C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#28042
C2×#24312
#165592
#163572
#158842
#156742
#154682
#153772
#150502
#147372
#144192
#139562
#138322